The n-Category Café

sticking point

Ah, right. My fault.

Actually, I don’t like my last proposal anymore for other reasons, too: if I am going to embed $\mathrm{DGCAs}\hookrightarrow \mathrm{somewhere}$ to use nice properties in $\mathrm{somewhere}$ I can just as well stick with my previous approach, which used $\mathrm{DGCAs}\hookrightarrow {\mathrm{Set}}^{S^{\mathrm{op}}}$ and then the internal hom in ${\mathrm{Set}}^{S^{\mathrm{op}}}$.

[Recall, above and a few weeks back, we (Todd Trimble and I, mostly) were trying to find the right generalization of “$C^{\mathrm{\infty }}$-algebras” from Moerdijk-Reyes’ book, which are product-preserving co-presheaves on smooth test domains to “$C^{\mathrm{\infty }}\mathrm{DGCAs}$”: $C^{\mathrm{\infty }}$-differential graded commutative algebras. ]

I am currently thinking about this approach:

a $C^{\mathrm{\infty }}$DGCA should be at least a DGCA whose degree 0 part is a $C^{\mathrm{\infty }}$-algebra. The idea is: this characterization is already a complete characterization.

So:

We have a forgetful functor

$${\mathrm{ev}}_{ℝ}:C^{\mathrm{\infty }}-\mathrm{Algebras}\to \mathrm{CommAlg}$$

which sends each $C^{\mathrm{\infty }}$-algebra $A$ to its “underlying algebra”

$$A\mapsto A(ℝ)$$

and the functor

$$(\cdot {)}^0:\mathrm{DGCAs}\to \mathrm{CommAlg}$$

which sends each DGCA $A$ to its degree 0 part

$$A\mapsto A^0.$$

proposed definition: The category $$C^{\mathrm{\infty }}\mathrm{DGCAs}$$ of $C^{\mathrm{\infty }}\mathrm{DGCAs}$ is the pullback $$\begin{array}{ccc}{C}^{\mathrm{\infty }}\mathrm{DGCAs}& \to & \mathrm{DGCAs}\\ \downarrow & & {\downarrow }^{(\cdot {)}^0}\\ C^{\mathrm{\infty }}-\mathrm{Algebras}& \stackrel{{\mathrm{ev}}_{ℝ}}{\to }& \mathrm{CommAlg}\end{array}$$

So a $C^{\mathrm{\infty }}\mathrm{DGCA}$ is a pair, consisting of an ordinary DGCA $A$ and a $C^{\mathrm{\infty }}$-algebra $A(\cdot )$, such that there is an isomorphism of commutative algebras $$A^0\simeq A(ℝ).$$

Notice that, in particular, ordinary algebras of smooth differential forms are $C^{\mathrm{\infty }}-\mathrm{DGCAs}$ in the obvious way:

$${\Omega }^{•}({ℝ}^n)\in C^{\mathrm{\infty }}\mathrm{DGCAs}.$$

Since underlying any morphism of $\mathrm{DGCAs}$ is always a morphism of the corresponding commutative algebras in degree 0, we are bound to inherit much of the nice structure on $C^{\mathrm{\infty }}-\mathrm{Algebras}$ this way.

In particular, it is easy to see that the coproduct

$${\otimes }_{\mathrm{\infty }}$$

on $C^{\mathrm{\infty }}-\mathrm{DGCAs}$ is automatically the “completed tensor product” in that

$${\Omega }^{•}({ℝ}^n){\otimes }_{\mathrm{\infty }}{\Omega }^{•}({ℝ}^m)\simeq {\Omega }^{•}({ℝ}^n\times {ℝ}^m).$$

This follows from the corresponding fact for the underlying degree 0 part

$${\Omega }^0({ℝ}^n)=C^{\mathrm{\infty }}({ℝ}^n)$$

and the obervation on p. 22 of Moerdijk-Reyes (which in turn is a consequence of their proposition 1.1 that $C^{\mathrm{\infty }}({ℝ}^n)$ is the free $C^{\mathrm{\infty }}$-algebra on $n$-generators and the universal property of the coproduct):

$${\Omega }^0({ℝ}^n){\otimes }_{\mathrm{\infty }}{\Omega }^0({ℝ}^m)\simeq {\Omega }^0({ℝ}^n\times {ℝ}^m).$$

From this the statement follows using the fact that a DGCA morphism

$${\Omega }^{•}(X)\to {\Omega }^{•}(Y)$$

is entirely fixed by its action on the degree 0-part, because the rest follows from the fact that differential forms can locally be written as

$$\omega ={\omega }_0\wedge d{\omega }_1\wedge d{\omega }_2\wedge \cdots \wedge d{\omega }_n$$

with ${\omega }_i\in C^{\mathrm{\infty }}(X)$

and the fact that a DGCA morphism has to commute with differentials:

$$\varphi (\omega )=\varphi ({\omega }_0)\wedge \varphi (d{\omega }_1)\wedge \varphi (d{\omega }_2)\wedge \cdots \wedge \varphi (d{\omega }_n)$$ $$=\varphi ({\omega }_0)\wedge d\varphi ({\omega }_1)\wedge d\varphi ({\omega }_2)\wedge \cdots \wedge d\varphi ({\omega }_n).$$

So that’s a first important consistency check that the definition of $C^{\mathrm{\infty }}\mathrm{DGCAs}$ above makes good sense.

Todd, I’m fine with ambimorphic. I prefer it to janusian, about which I share your reservations. Actually, I don’t think I’d ever heard the word ambimorphic before you suggested it, so it’s completely neutral for me — except on odd occasions when it makes me think of frogs and newts.

Part of the reason why I’ve been quiet about this is that I’m aware of not being very good at choosing terminology. I only know which word I don’t like.

OK, I’m with you.

Even overlooking the use of the word in the spoof paper A Simple Proof of the Riemann Hypothesis by Leon Oiler, John Grauss, Joe Lagrunge, John Dirishlay, and Joe Fouray:

Sasaki constructed several ambimorphic solutions,

and the mention of ‘Journal of Ambimorphic Information’ in the bibliography.

Someone evidently finds the name amusing.

Well, as I say I don’t think we have to decide right away. What I’ll do is try ‘ambimorphic’ on for size for myself, and if other people like and start using it too, we can (for the purposes of this blog) stick with it. If at this point someone prefers ‘janusian’, he or she can either declare that, or [more quietly and surreptitiously] slip in the usage in a comment somewhere when the opportunity presents itself. Or suggest something else.

But (sad to say), my prediction is that none of this will matter anyway: people will continue using ‘schizophrenic’, despite Tom’s valid objections. Established terminology in mathematics has a terrible inertia: it would take a lot of momentum to get enough people to be bothered to say, “ambimorphic, the-concept-formerly-known-as schizophrenic”. (What do people call Prince these days, anyway?)

Toby Bartels [who has apparently stopped commenting here]

I’ve even stopped regularly reading, but every once in a while I search for my name here.

Anyway, I just want to say that I like the term “surgery” too!

I’ll be just shy of absolutely precise. The way I’ve generally seen it is that an ambimorphic structure consists of two structures on the same underlying object $S$ [let’s say a set for now] which “commute” with one another, say a $C$-structure and a $D$-structure, such that the lift

of ${\mathrm{hom}}_C(-,S):C^{\mathrm{op}}\to \mathrm{Set}$, and the lift

of ${\mathrm{hom}}_D(-,S):D^{\mathrm{op}}\to \mathrm{Set}$, are adjoint in the appropriate sense. (Most of the time where I’ve seen the concept applied, this adjunction is actually an equivalence between $C^{\mathrm{op}}$ and $D$.)

A classical example is Stone duality. The set $2$ carries both a Boolean algebra structure and a structure of Stone space (a compact Hausdorff totally disconnected space), in such a way that it induces a contravariant equivalence between the category of Boolean algebras $C$ and the category of Stone spaces $D$.